Biconvex optimization

Biconvex optimization is a generalization of convex optimization where the objective function and the constraint set can be biconvex.

There are methods that can find the global optimum of these problems.

[1][2] A set

is called a biconvex set on

is a convex set in

is a convex set in

A function

is called a biconvex function if fixing

A common practice for solving a biconvex problem (which does not guarantee global optimality of the solution) is alternatively updating

by fixing one of them and solving the corresponding convex optimization problem.

[1] The generalization to functions of more than two arguments is called a block multi-convex function.

is block multi-convex iff it is convex with respect to each of the individual arguments while holding all others fixed.

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